In such a setting the unitary evolution is found
generated by a non-Hermitian Hamiltonian \(H\)  because
such an operator (called “quasi-Hermitian” [1]}
alias  “crypto-Hermitian” [2]})
appears Hermitizable.
In applications the Hermitization
has to be visualized as a
constructive
transfer of representation of the quantum bound
states of interest from the auxiliary Hilbert space \({\cal H}_{math}\) 
to the traditional physical Hilbert space \({\cal L}\)  of textbooks
or, better [3]},
to one of its other suitable representations (say, \({\cal H}_{phys}\) ).
